Recursive estimation of states of constrained nonlinear dynamic systems has attracted the attention of many researchers in recent years. In this work, we propose a constrained recursive formulation of the ensemble Kalman filter (EnKF) that retains the advantages of the unconstrained EnKF while, systematically dealing with bounds on the estimated states. The EnKF belongs to the class of particle filters that are increasingly being used for solving state estimation problems associated with nonlinear systems. A highlight of our approach is the use of truncated multivariate distributions for systematically solving the estimation problem in the presence of state constraints. The efficacy of the proposed constrained state estimation algorithm using the EnKF is illustrated by application on two benchmark problems in the literature (a simulated gas-phase reactor and an isothermal batch reactor) involving constraints on estimated state variables and another example problem, which involves constraints on the process noise.